Arcs A

An arc is part of the circumference of a circle.

Length of arc = θ

x 2πr 360

Sectors A The shaded area is a sector. It is bounded by an arc and two radii.

Area of sector = θ

x πr2

360

Sectors and arcs26.1

82 Geometry and algebra Arcs, cones and spheres 83

Calculate the missing lengths in these circles. Give your answers in terms of π where necessary.

b c d

Calculate the area of each sector. Give your answers in terms of π.

a b c d

Andrea wants to plant radishes. Each radish plant needs an area of 30 cm2 to grow. The diagram shows one of Andrea’s vegetable beds. It is made from a rectangle and a sector of a circle. How many radish plants can Andrea grow in this vegetable bed?

What percentage of this circle is shaded? Give your answer to 3 s.f.

Calculate the volume and surface area of each cone. Give your answers correct to 3 s.f.

a b c d

A cone has volume 550 cm3. If the height of the cone is 15 cm work out correct to 3 s.f.:

a the radius of the cone

b the slant height of the cone

c the total surface area of the cone.

Karl makes a horizontal cut half way up this cone. Find the ratio of the volume of the smaller piece to the larger piece.

A metal cube of side length 8 cm is melted down. The metal is used to make a cone with radius 6 cm. Calculate the height of the cone correct to 3 s.f.

A plinth for a statue is made by cutting the top off of a cone. Calculate the volume of the plinth. Give your answer correct to 2 d.p.

Nisha cuts this sector out of cardboard and folds it up to make a cone-shaped cup. What is the volume of Nisha’s cup?

Arcs, cones and spheres26Cones26.2

Links to:Higher Student Book

Ch.26, pp.521–527

A

AAO2

AO3

A*

A

A*

A*

A

AO2

AO3

AO2

1

2

3

4 5

Key Points

Cones A A*

Volume of cone = 1 πr2h

3

Curved surface area of cone = πrl

Spheres A A*

Volume of sphere = 4 πr3

3

Surface area of sphere = 4 πr2

1

2

3

4

5

6

60̊

150̊

x

6 cm

306̊144̊

2π cm

9π cmx280̊

x

θ

Arc length

rθ

r

r

hr

l

4 cm

10 cm 2 mm

3 m

4 m

5 m

9 mm

10 m 12 m3.8 cm

5 cm

18mm

x

120̊ 6 cm

75̊64̊

40̊

50 cm

20 cm

72̊ 40 mm

14 mm

r

135̊

20 cm

8 cm

θ

α

x

x

60̊

150̊

x

6 cm

306̊144̊

2π cm

9π cmx280̊

x

θ

Arc length

rθ

r

r

hr

l

4 cm

10 cm 2 mm

3 m

4 m

5 m

9 mm

10 m 12 m3.8 cm

5 cm

18mm

x

120̊ 6 cm

75̊64̊

40̊

50 cm

20 cm

72̊ 40 mm

14 mm

r

135̊

20 cm

8 cm

θ

α

x

x

60̊

150̊

x

6 cm

306̊144̊

2π cm

9π cmx280̊

x

θ

Arc length

rθ

r

r

hr

l

4 cm

10 cm 2 mm

3 m

4 m

5 m

9 mm

10 m 12 m3.8 cm

5 cm

18mm

x

120̊ 6 cm

75̊64̊

40̊

50 cm

20 cm

72̊ 40 mm

14 mm

r

135̊

20 cm

8 cm

θ

α

x

x

60̊

150̊

x

6 cm

306̊144̊

2π cm

9π cmx280̊

x

θ

Arc length

rθ

r

r

hr

l

4 cm

10 cm 2 mm

3 m

4 m

5 m

9 mm

10 m 12 m3.8 cm

5 cm

18mm

x

120̊ 6 cm

75̊64̊

40̊

50 cm

20 cm

72̊ 40 mm

14 mm

r

135̊

20 cm

8 cm

θ

α

x

x

60̊

150̊

x

6 cm

306̊144̊

2π cm

9π cmx280̊

x

θ

Arc length

rθ

r

r

hr

l

4 cm

10 cm 2 mm

3 m

4 m

5 m

9 mm

10 m 12 m3.8 cm

5 cm

18mm

x

120̊ 6 cm

75̊64̊

40̊

50 cm

20 cm

72̊ 40 mm

14 mm

r

135̊

20 cm

8 cm

θ

α

x

x

60̊

150̊

x

6 cm

306̊144̊

2π cm

9π cmx280̊

x

θ

Arc length

rθ

r

r

hr

l

4 cm

10 cm 2 mm

3 m

4 m

5 m

9 mm

10 m 12 m3.8 cm

5 cm

18mm

x

120̊ 6 cm

75̊64̊

40̊

50 cm

20 cm

72̊ 40 mm

14 mm

r

135̊

20 cm

8 cm

θ

α

x

x

60̊

150̊

x

6 cm

306̊144̊

2π cm

9π cmx280̊

x

θ

Arc length

rθ

r

r

hr

l

4 cm

10 cm 2 mm

3 m

4 m

5 m

9 mm

10 m 12 m3.8 cm

5 cm

18mm

x

120̊ 6 cm

75̊64̊

40̊

50 cm

20 cm

72̊ 40 mm

14 mm

r

135̊

20 cm

8 cm

θ

α

x

x

60̊

150̊

x

6 cm

306̊144̊

2π cm

9π cmx280̊

x

θ

Arc length

rθ

r

r

hr

l

4 cm

10 cm 2 mm

3 m

4 m

5 m

9 mm

10 m 12 m3.8 cm

5 cm

18mm

x

120̊ 6 cm

75̊64̊

40̊

50 cm

20 cm

72̊ 40 mm

14 mm

r

135̊

20 cm

8 cm

θ

α

x

x

60̊

150̊

x

6 cm

306̊144̊

2π cm

9π cmx280̊

x

θ

Arc length

rθ

r

r

hr

l

4 cm

10 cm 2 mm

3 m

4 m

5 m

9 mm

10 m 12 m3.8 cm

5 cm

18mm

x

120̊ 6 cm

75̊64̊

40̊

50 cm

20 cm

72̊ 40 mm

14 mm

r

135̊

20 cm

8 cm

θ

α

x

x

60̊

150̊

x

6 cm

306̊144̊

2π cm

9π cmx280̊

x

θ

Arc length

rθ

r

r

hr

l

4 cm

10 cm 2 mm

3 m

4 m

5 m

9 mm

10 m 12 m3.8 cm

5 cm

18mm

x

120̊ 6 cm

75̊64̊

40̊

50 cm

20 cm

72̊ 40 mm

14 mm

r

135̊

20 cm

8 cm

θ

α

x

x

60̊

150̊

x

6 cm

306̊144̊

2π cm

9π cmx280̊

x

θ

Arc length

rθ

r

r

hr

l

4 cm

10 cm 2 mm

3 m

4 m

5 m

9 mm

10 m 12 m3.8 cm

5 cm

18mm

x

120̊ 6 cm

75̊64̊

40̊

50 cm

20 cm

72̊ 40 mm

14 mm

r

135̊

20 cm

8 cm

θ

α

x

x

You need to remember the formulae for arcs and sectors in your exam.

The formulae for cones and spheres are given on your exam paper.

Use similar triangles to work out the height of the original cone.

This earring is made from a piece of wire of length 58 mm. Find the length r.

60̊

150̊

x

6 cm

306̊144̊

2π cm

9π cmx280̊

x

θ

Arc length

rθ

r

r

hr

l

4 cm

10 cm 2 mm

3 m

4 m

5 m

9 mm

10 m 12 m3.8 cm

5 cm

18mm

x

120̊ 6 cm

75̊64̊

40̊

50 cm

20 cm

72̊ 40 mm

14 mm

r

135̊

20 cm

8 cm

θ

α

x

x

Links to:Section 26.3, p.521

Links to:Section 26.5, p.526

60̊

150̊

x

6 cm

306̊144̊

2π cm

9π cmx280̊

x

θ

Arc length

rθ

r

r

hr

l

4 cm

10 cm 2 mm

3 m

4 m

5 m

9 mm

10 m 12 m3.8 cm

5 cm

18mm

x

120̊ 6 cm

75̊64̊

40̊

50 cm

20 cm

72̊ 40 mm

14 mm

r

135̊

20 cm

8 cm

θ

α

x

x

Challenge yourself You won’t encounter questions like this in the exam, but the underlying maths is covered in your GCSE course. Have a go!

This sector can be folded up to make a cone.

The angle in the sector is θ. The angle that the curved surface of the finished cone makes with the horizontal is α.

Investigate the relationship between θ and α. Do you need to know the radius of the sector to calculate α? Can you find a formula for α in terms of θ?

60̊

150̊

x

6 cm

306̊144̊

2π cm

9π cmx280̊

x

θ

Arc length

rθ

r

r

hr

l

4 cm

10 cm 2 mm

3 m

4 m

5 m

9 mm

10 m 12 m3.8 cm

5 cm

18mm

x

120̊ 6 cm

75̊64̊

40̊

50 cm

20 cm

72̊ 40 mm

14 mm

r

135̊

20 cm

8 cm

θ

α

x

x

60̊

150̊

x

6 cm

306̊144̊

2π cm

9π cmx280̊

x

θ

Arc length

rθ

r

r

hr

l

4 cm

10 cm 2 mm

3 m

4 m

5 m

9 mm

10 m 12 m3.8 cm

5 cm

18mm

x

120̊ 6 cm

75̊64̊

40̊

50 cm

20 cm

72̊ 40 mm

14 mm

r

135̊

20 cm

8 cm

θ

α

x

x

You can use trigonometry to help with this investigation. See Chapter 23 for a reminder about sine, cosine and tangent.